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Author: Leon Horsten
Publisher: Oxford University Press
ISBN: 0198759592
Category : Mathematics
Languages : en
Pages : 289
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Book Description
The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.
Author: Leon Horsten
Publisher: Oxford University Press
ISBN: 0198759592
Category : Mathematics
Languages : en
Pages : 289
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Book Description
The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.
Author: Kurt Gödel
Publisher: Oxford University Press, USA
ISBN: 0195072553
Category : Mathematics
Languages : en
Pages : 558
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Book Description
"Anyone interested in the life and work of Kurt Gödel, or in the history of mathematical logic in this century, is indebted to all of the contributors to this volume for the care with which they have presented Gödel's work. They have succeeded in using their own expertise to elucidate both the nature and significance of what Gödel and, in turn, mathematical logic have accomplished." --Isis (on volume I). The third volume brings togetherGödels unpublished essays and lectures.
Author: Jean van Heijenoort
Publisher: Harvard University Press
ISBN: 0674257243
Category : Philosophy
Languages : en
Pages : 684
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Book Description
The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege’s Begriffsschrift that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory. Frege’s book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to Principia Mathematica. Burali-Forti, Cantor, Russell, Richard, and König mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Löwenheim’s theorem, and he and Fraenkel amend Zermelo’s axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Gödel, including the latter’s famous incompleteness paper. Of the forty-five contributions here collected all but five are presented in extenso. Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.
Author: Juliette Kennedy
Publisher: Cambridge University Press
ISBN: 1139991752
Category : Science
Languages : en
Pages : 293
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Book Description
The logician Kurt Gödel (1906–1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.
Author: Carl Posy
Publisher: Springer Nature
ISBN: 3031216555
Category : Mathematics
Languages : en
Pages : 404
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Book Description
This book provides a survey of a number of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the papers in this volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike.
Author: Richard L. Tieszen
Publisher: Oxford University Press
ISBN: 019960620X
Category : Biography & Autobiography
Languages : en
Pages : 258
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Book Description
Richard Tieszen analyzes, develops, and defends the writings of Kurt Gödel (1906-1978) on the philosophy and foundations of mathematics and logic. Gödel's relation to the work of Plato, Leibniz, Husserl, and Kant is examined, and a new type of platonic rationalism that requires rational intuition, called 'constituted platonism', is proposed.
Author: Jeff Buechner
Publisher: MIT Press
ISBN: 0262261979
Category : Philosophy
Languages : en
Pages : 357
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Book Description
The first systematic examination of Hilary Putnam's arguments against computational functionalism challenges each of Putnam's main arguments. With mind-brain identity theories no longer dominant in philosophy of mind in the late 1950s, scientific materialists turned to functionalism, the view that the identity of any mental state depends on its function in the cognitive system of which it is a part. The philosopher Hilary Putnam was one of the primary architects of functionalism and was the first to propose computational functionalism, which views the human mind as a computer or an information processor. But, in the early 1970s, Putnam began to have doubts about functionalism, and in his masterwork Representation and Reality (MIT Press, 1988), he advanced four powerful arguments against his own doctrine of computational functionalism. In Gödel, Putnam, and Functionalism, Jeff Buechner systematically examines Putnam's arguments against functionalism and contends that they are unsuccessful. Putnam's first argument uses Gödel's incompleteness theorem to refute the view that there is a computational description of human reasoning and rationality; his second, the “triviality argument,” demonstrates that any computational description can be attributed to any physical system; his third, the multirealization argument, shows that there are infinitely many computational realizations of an arbitrary intentional state; his fourth argument buttresses this assertion by showing that there cannot be local computational reductions because there is no computable partitioning of the infinity of computational realizations of an arbitrary intentional state into a single package or small set of packages (equivalence classes). Buechner analyzes these arguments and the important inferential connections among them—for example, the use of both the Gödel and triviality arguments in the argument against local computational reductions—and argues that none of Putnam's four arguments succeeds in refuting functionalism. Gödel, Putnam, and Functionalism will inspire renewed discussion of Putnam's influential book and will confirm Representation and Reality as a major work by a major philosopher.
Author: Mario Alai
Publisher: Springer
ISBN: 3319163698
Category : Science
Languages : en
Pages : 334
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Book Description
This book offers the most complete and up-to-date overview of the philosophical work of Evandro Agazzi, presently the most important Italian philosopher of science and one of the most influential in the world. Scholars from seven countries explore his contributions in areas ranging from philosophy of physics and general philosophy of science to bioethics, philosophy of mathematics and logic, epistemology of the social sciences and history of science, philosophy of language and artificial intelligence, education and anthropology, metaphysics and philosophy of religion. Agazzi developed a complete and coherent philosophical system, anticipating some of the turns in the philosophy of science after the crisis of logical empiricism and exerting an equal influence on continental hermeneutic philosophy. His work is characterized by an original synthesis of contemporary analytic philosophy, phenomenology and classical philosophy, including the scholastic tradition and these threads are reflected in the different backgrounds of the contributors to this book. While upholding the epistemological value of science against scepticism and relativism, Agazzi eschews scientism by stressing the equal importance of non-scientific forms of thought, such as metaphysics and religion. While defending the freedom of research as a cognitive enterprise, he argues that as a human and social practice it must nonetheless respect ethical constraints.
Author: Mario Piazza
Publisher: Springer
ISBN: 3319933426
Category : Mathematics
Languages : en
Pages : 278
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Book Description
This book contains more than 15 essays that explore issues in truth, existence, and explanation. It features cutting-edge research in the philosophy of mathematics and logic. Renowned philosophers, mathematicians, and younger scholars provide an insightful contribution to the lively debate in this interdisciplinary field of inquiry. The essays look at realism vs. anti-realism as well as inflationary vs. deflationary theories of truth. The contributors also consider mathematical fictionalism, structuralism, the nature and role of axioms, constructive existence, and generality. In addition, coverage also looks at the explanatory role of mathematics and the philosophical relevance of mathematical explanation. The book will appeal to a broad mathematical and philosophical audience. It contains work from FilMat, the Italian Network for the Philosophy of Mathematics. These papers collected here were also presented at their second international conference, held at the University of Chieti-Pescara, May 2016.
Author: Maria Hämeen-Anttila
Publisher: Springer Nature
ISBN: 303137875X
Category : Mathematics
Languages : en
Pages : 327
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Book Description
Kurt Gödel (1906-1978) gained world-wide fame by his incompleteness theorem of 1931. Later, he set as his aim to solve what are known as Hilbert's first and second problems, namely Cantor's continuum hypothesis about the cardinality of real numbers, and secondly the consistency of the theory of real numbers and functions. By 1940, he was halfway through the first problem, in what was his last published result in logic and foundations. His intense attempts thereafter at solving these two problems have remained behind the veil of a forgotten German shorthand he used in all of his writing. Results on Foundations is a set of four shorthand notebooks written in 1940-42 that collect results Gödel considered finished. Its main topic is set theory in which Gödel anticipated several decades of development. Secondly, Gödel completed his 1933 program of establishing the connections between intuitionistic and modal logic, by methods and results that today are at the same time new and 80 years old. The present edition of Gödel's four notebooks encompasses the 368 numbered pages and 126 numbered theorems of the Results on Foundations, together with a list of 74 problems on set theory Gödel prepared in 1946, and a list of an unknown date titled "The grand program of my research in ca. hundred questions.''
